Machine Kinematics
solved, then drawn
Concept

Denavit–Hartenberg parameters — where it appears

A description of a chain by four numbers a joint, derived from the common perpendicular between consecutive axes and undefined when they are parallel. It is the classical description and its weakness is geometric rather than numerical: the common perpendicular between two parallel axes is not unique, so the parameters are not either.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

SCARA, as four numbers a joint. A Denavit–Hartenberg table describes each joint by the common perpendicular between its axis and the next one: how long it is, how much the axes twist across it, where along the first axis it meets, and at what angle. 3 of these 3 rows have no answer for the last two, because the axes they describe are parallel and two parallel lines have infinitely many common perpendiculars, all the same length, at every point along them. The arm is perfectly ordinary; it is the description that has run out.

Four numbers or a screw

An arm can be written down as four numbers a joint or as a line in space with a pitch on it. Both are minimal, both describe the same machine to the last bit, and one of them jumps by three hundred and fifty thousand when an axis is tilted by a millionth of a radian.

serial · Serial Bennett's four-bar at 40°. Four bars, four revolute joints, and axes that are not parallel — a spatial four-bar, which Kutzbach counts at -2 degrees of freedom. Bennett's condition, sin α / a = sin β / b, makes the screw system rank 3 instead of 4, so the mechanism has 1. Driving the first joint through a full turn, 48 of 48 positions assemble. Orthographic projection, viewed from 40° azimuth and 24° elevation; dashed stubs mark the joint axes.

Bennett, and the condition that moves it

A spatial four-bar is immobile by every count there is, and generically it cannot even be assembled at more than isolated configurations. Bennett found the one relation between four lengths and two twists that makes it turn through a full revolution — and break the relation by two parts in a thousand and most of the travel is gone.

spatial · Spatial What a calibration can and cannot see. The singular values of elbow arm's identification Jacobian — the matrix of how the tool pose moves when each model parameter is nudged, over 11 postures. There are 36 parameters and only 30 of them can be found: the last 6 directions come out at 2.3e-8, which is the difference noise, against 6.3e-4 for the weakest real one — a gap of 3e+4. And 30 is exactly 4R + 2P + 6 for this arm's 6 turning and 0 sliding joints, which is a count from the literature meeting a rank measured from the arm's own arithmetic.

What a calibration cannot see

A six-joint arm's model has thirty-six parameters and a measurement can find thirty of them. The other six are not hard to measure — they are combinations that move the tool by exactly nothing, at every posture, and no instrument ever built will separate them. The count is 4R + 2P + 6, and it comes out of a rank on four different arms.

serial · Serial

Named alongside it

The objects these essays reach for when they reach for this one.

ConditioningParameterisationRankScrewSerial manipulatorTwistBennett's linkageCalibrationCommon normalConstraintForward kinematicsIdentification

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